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# A Closed Form for Slant Submanifolds of Generalized Sasakian Space Forms

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## Abstract

The Maslov form is a closed form for a Lagrangian submanifold of C m , and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisfies a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal.
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... For a slant submanifold of a generalized Sasakian manifold the Maslov form is not always closed. At [5] we present a form that is always closed for a slant submanifold, so it really plays the role of the Maslov form in the cited papers. However, in the case where the submanifold satisfies the equality in the above inequality, such form is not always a conformal vector field. ...
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This book contains an up-to-date survey and self-contained chapters on complex slant submanifolds and geometry, authored by internationally renowned researchers. The book discusses a wide range of topics, including slant surfaces, slant submersions, nearly Kaehler, locally conformal Kaehler, and quaternion Kaehler manifolds. It provides several classification results of minimal slant surfaces, quasi-minimal slant surfaces, slant surfaces with parallel mean curvature vector, pseudo-umbilical slant surfaces, and biharmonic and quasi biharmonic slant surfaces in Lorentzian complex space forms. Furthermore, this book includes new results on slant submanifolds of para-Hermitian manifolds.
... For a slant submanifold of a generalized Sasakian manifold the Maslov form is not always closed. In [5] we present a form that is always closed for a slant submanifold, so it really plays the role of the Maslov form in the cited papers. However, in the case where the submanifold satisfies the equality in the above inequality, such form is not always conformal. ...
Book
Full-text available
The book gathers a wide range of topics such as warped product semi-slant submanifolds, slant submersions, semi-slant ξ^⊥-, hemi-slant ξ^⊥-Riemannian submersions, quasi hemi-slant submanifolds, slant submanifolds of metric f-manifolds, slant lightlike submanifolds, geometric inequalities for slant submanifolds, 3-slant submanifolds, and semi-slant submanifolds of almost paracontact manifolds. The book also includes interesting results on slant curves and magnetic curves, where the latter represents trajectories moving on a Riemannian manifold under the action of magnetic field. It presents detailed information on the most recent advances in the area, making it of much value to scientists, educators and graduate students.
Chapter
Slant submanifolds were defined in the nineties by B.-Y. Chen and the corresponding theory has had an increasing development.2000 AMS Mathematics Subject Classification53C4053C4253C50
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The present volume is the written version of a series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's recent work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Japan, Professor P. F. Leung of Singapore and Professor J. M. Morvan of France on geometry of slant submanifolds and its related subjects in a systematical way.
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